The aim of these lec­tures is to re­port on the pro­gress of the in­dex prob­lem in the last year. We will de­scribe an ex­ten­sion of the in­dex for­mula for closed man­i­folds (see [Atiyah and Sing­er 1963]) to man­i­folds with bound­ary. The work of Sec­tion 4, i.e., the proof of the gen­er­al in­dex the­or­em from The­or­em 1 was done in col­lab­or­a­tion with Sing­er.

The peri­od­icity the­or­em for the in­fin­ite unit­ary group [Bott 1959] can be in­ter­preted as a state­ment about com­plex vec­tor bundles. As such it de­scribes the re­la­tion between vec­tor bundles over \( X \) and \( X\times S^2 \), where \( X \) is a com­pact space and \( S^2 \) is the 2-sphere. This re­la­tion is most suc­cinctly ex­pressed by the for­mula
\[ K(X\times S^2) \simeq K(X)\otimes K(S^2), \]
where \( K(X) \) is the Grothen­dieck group of com­plex vec­tor bundles over \( X \). The gen­er­al the­ory of these \( K \)-groups, as de­veloped in [Atiyah and Hirzebruch 1961], has found many ap­plic­a­tions in to­po­logy and re­lated fields. Since the peri­od­icity the­or­em is the found­a­tion stone of all this the­ory it seems de­sir­able to have an ele­ment­ary proof of it, and it is the pur­pose of this pa­per to present such a proof.

The clas­sic­al Lef­schetz fixed point for­mula ex­presses, un­der suit­able cir­cum­stances, the num­ber of fixed points of a con­tinu­ous map \( f:X\to X \) in terms of the trans­form­a­tion in­duced by \( f \) on the co­homo­logy of \( X \). If \( X \) is not just a to­po­lo­gic­al space but has some fur­ther struc­ture, and if this struc­ture is pre­served by \( f \), one would ex­pect to be able to re­fine the Lef­schetz for­mula and to say more about the nature of the fixed points. The pur­pose of this note is to present such a re­fine­ment (The­or­em 1) when \( X \) is a com­pact dif­fer­en­ti­able man­i­fold en­dowed with an el­lipt­ic dif­fer­en­tial op­er­at­or (or more gen­er­ally an el­lipt­ic com­plex). Tak­ing es­sen­tially the clas­sic­al op­er­at­ors of com­plex and Rieman­ni­an geo­metry we ob­tain a num­ber of im­port­ant spe­cial cases (The­or­em 2, 3). The first of these was con­jec­tured to us by Shimura and was proved by Eichler for di­men­sion one.

Part I of this pa­per de­scribed an ex­ten­sion of the clas­sic­al Lef­sh­etz the­or­em in the frame­work of el­lipt­ic com­plexes. This second part is de­voted to the ap­plic­a­tions and ex­amples of this ex­ten­sion which for the most part were an­nounced in [Atiyah and Bott 1966].

The the­ory of la­cunas for hy­per­bol­ic dif­fer­en­tial op­er­at­ors was cre­ated by I. G. Pet­rovsky who pub­lished the ba­sic pa­per of the sub­ject in 1945. Al­though its res­ults are very clear, the pa­per is dif­fi­cult read­ing and has so far not lead to stud­ies of the same scope. We shall cla­ri­fy and gen­er­al­ize Pet­rovsky’s the­ory.

The joint pa­per of the above title which ap­peared in In­ven­tiones Math.19, 279–330 (1973), though cor­rect in prin­ciple, con­tained some tech­nic­al er­rors which we shall here ex­plain and rec­ti­fy. Our thanks are due to D. Ep­stein, Y. Colin de Ver­diére and A. Vasquez whose com­pu­ta­tions and quer­ies aler­ted us to our er­rors.

The Yang–Mills func­tion­al over a Riemann sur­face is stud­ied from the point of view of Morse the­ory. The main res­ult is that this is a ‘per­fect’ func­tion­al provided due ac­count is taken of its gauge sym­metry. This en­ables to­po­lo­gic­al con­clu­sions to be drawn about the crit­ic­al sets and leads even­tu­ally to in­form­a­tion about the mod­uli space of al­geb­ra­ic bundles over the Riemann sur­face. This in turn de­pends on the in­ter­play between the holo­morph­ic and unit­ary struc­tures, which is ana­lysed in de­tail.

The pur­pose of this note is to present a de Rham ver­sion of the loc­al­iz­a­tion the­or­ems of equivari­ant co­homo­logy, and to point out their re­la­tion to a re­cent res­ult of Duister­maat and Heck­man and also to a quite in­de­pend­ent res­ult of Wit­ten. To a large ex­tent all the ma­ter­i­al that we use has been around for some time, al­though equivari­ant co­homo­logy is not per­haps fa­mil­i­ar to ana­lysts. Our con­tri­bu­tion is there­fore mainly an ex­pos­it­ory one link­ing to­geth­er vari­ous points of view.

P. God­dard:
“Sir Mi­chael Atiyah and the early days of the New­ton In­sti­tute,”
pp. 23–​28
in
The founders of in­dex the­ory: Re­min­is­cences of and about Sir Mi­chael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Sing­er.
Edi­ted by S.-T. Yau.
In­ter­na­tion­al Press (Somerville, MA),
2003.
Also in the 2009 edi­tion of the book.incollection

The first part of this es­say com­prises some brief re­min­is­cences from my time as a re­search stu­dent of Sir Mi­chael Atiyah: these will be com­mon­place to my con­tem­por­ar­ies, but per­haps young­er math­em­aticians may be less fa­mil­i­ar with the re­search in­terests of this peri­od. In the second part of the es­say I will dis­cuss some cur­rent re­search ques­tions.

The Bib­li­o­graph­ic Data, be­ing a mat­ter of fact and
not cre­at­ive ex­pres­sion, is not sub­ject to copy­right.
To the ex­tent pos­sible un­der law,
Math­em­at­ic­al Sci­ences Pub­lish­ers
has waived all copy­right and re­lated or neigh­bor­ing rights to the
Bib­li­o­graph­ies on Cel­eb­ra­tio Math­em­at­ica,
in their par­tic­u­lar ex­pres­sion as text, HTML, Bib­TeX data or oth­er­wise.

The Ab­stracts of the bib­li­o­graph­ic items may be copy­righted ma­ter­i­al whose use has not
been spe­cific­ally au­thor­ized by the copy­right own­er.
We be­lieve that this not-for-profit, edu­ca­tion­al use con­sti­tutes a fair use of the
copy­righted ma­ter­i­al,
as provided for in Sec­tion 107 of the U.S. Copy­right Law. If you wish to use this copy­righted ma­ter­i­al for
pur­poses that go bey­ond fair use, you must ob­tain per­mis­sion from the copy­right own­er.